Invariants of quartic plane curves as automorphic forms

Abstract

ABSTRACT. We identify the algebra of regular functions on the space of quartic polynomials in three complex variables invariant under SL(3, C) with an algebra of meromorphic automorphic forms on the complex 6-ball. We also discuss the underlying geometry.

Computing Moment Maps of Hypersurfaces using MAXIMA

“…With these preparation, we construct the moment map of a general complex polynomial C:lsum((Re[Wt(j) [1],Wt(j) [2],Wt(j) [3]] +%i*Im[Wt(j) [1],Wt(j) [2],Wt(j) [3]])*j,j,M);…”

“…Then the hermitian product F, G is conjugate(WtCoe(F)).WtCoe(G); Then m is the moment matrix of C, the expression Lm is the square length m 2 of m. Let R be the general real polynomial R:lsum(a[Wt(j) [1],Wt(j) [2],Wt(j) [3]]*j,j,M(d)); a 0,0,3 z 3 + a 0,1,2 yz 2 + a 1,0,2 xz 2 + a 0,2,1 y 2 z + a 1,1,1 xyz + a 2,0,1 x 2 z + a 0,3,0 y 3 + a 1,2,0 xy 2 + a 2,1,0 x 2 y + a 3,0,0 x 3 then by replacing C into R, the construction above gives the moment matrix m of R and the square length m 2 of R.…”

Centro-Equiaffine Differential Invariants of Curve Families

“…The first comprehensive treatment of affine geometry is given in the seminal work of Paukowitsch [13]. For further developments of subject, we refer the reader to [7], and more modern texts [1], [15], the commentaries [5], [10], [19] and survey papers [11], [3]. The fundamental theorem of curves in centro-affine geometry is obtained in [2].…”

Moduli spaces and locally symmetric varieties

“…It seems likely that we can obtain a bounded symmetric domain of type E 6 in X as a locus with additional symmetry. For instance if we have a s ∈ S such that h is nonzero on F 3 s and of signature (11,16) in F 2 s /F 3 s , then G s acts on H 2,1 s := F 2 s ∩ (F 3 s ) ⊥ through a real group of type E 6 and I suspect that for the correct sign of h on F 3 s , X ∩ P(F 2 s ) is its symmetric domain.…”

“…It seems likely that we can obtain a bounded symmetric domain of type E 6 in X as a locus with additional symmetry. For instance if we have a s ∈ S such that h is nonzero on F 3 s and of signature (11,16) in…”